Question

1. \\(\dfrac{6x^{3}y^{-7}}{9x^{2}y^{6}} =\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients \( \frac{6}{9} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 3. So \( \frac{6}{9}=\frac{6\div3}{9\div3}=\frac{2}{3} \).

Step2: Simplify the \( x \)-terms

For the \( x \)-terms, we use the rule of exponents \( \frac{x^m}{x^n}=x^{m - n} \). Here, \( m = 3 \) and \( n = 2 \), so \( \frac{x^3}{x^2}=x^{3-2}=x^1 = x \).

Step3: Simplify the \( y \)-terms

For the \( y \)-terms, we use the rule of exponents \( \frac{y^m}{y^n}=y^{m - n} \). Here, \( m=-7 \) and \( n = 6 \), so \( \frac{y^{-7}}{y^6}=y^{-7-6}=y^{-13} \). And we know that \( y^{-13}=\frac{1}{y^{13}} \) (by the rule \( a^{-n}=\frac{1}{a^n} \)).

Step4: Combine all the simplified terms

Multiply the simplified coefficient, \( x \)-term, and \( y \)-term together. So we have \( \frac{2}{3}\times x\times\frac{1}{y^{13}}=\frac{2x}{3y^{13}} \).

Answer:

\( \frac{2x}{3y^{13}} \)